Cohomology and Extensions
denotes the category of abelian groups: it has abelian groups as objects and group homomorphisms between these as morphisms.
The archetypical example of an abelian group is the group of integers, and for many purposes it is useful to think of equivalently as the category of modules over
The category serves as the basic enriching category in homological algebra. There Ab-enriched categories play much the same role as Set-enriched categories (locally small categories) play in general.
In this vein, the analog of in homotopy theory – or rather in stable homotopy theory – is the category of spectra, either regarded as the stable homotopy category or rather refined to the stable (infinity,1)-category of spectra. A spectrum is much like an abelian group up to coherent homotopy and the role of the archetypical abelian group is the played by the sphere spectrum
Free abelian groups
Direct sum, direct product and tensor product
We discuss basic properties of binary operations on the category of abelian groups: direct product, direct sum and tensor product. Below in Monoidal and bimonoidal structure we put these structures into a more abstract context.
For two abelian groups, their direct product is the abelian group whose elements are pairs with and , whose 0-element is and whose addition operation is the componentwise addition
This is at the same time the direct sum .
Similarly for FinSet Set a finite set, we have
But for a set which is not finite, there is a difference: the direct sum of an -indexed family of abelian groups is the sub-group of the direct product on those elements for which only finitely many components are non-0
The trivial group (the group with a single element) is a unit for the direct sum: for every abelian group we have
In view of remark 1 this means that the direct sum of copies of the additive group of integers with themselves is equivalently the free abelian group on :
See at tensor product of abelian groups for details.
The unit for the tensor product of abelian groups is the additive group of integers:
For and , the direct sum of copies of with itself is equivalently the tensor product of abelian groups of the free abelian group on with :
Monoidal and bimonoidal structure
With the definitions and properties discussed above in Direct sum, etc. we have the following
The category becomes a monoidal category
under direct sum ;
under tensor product of abelian groups .
Indeed with both structures combined we have
is a bimonoidal category (and can be made a bipermutative category).
Categories enriched over are called pre-additive categories or sometimes just additive categories. If they satisfy an extra exactness condition they are called abelian categories. See at additive and abelian categories.